Search & Filtering
Determine the set of interior points, exterior points,
accumulation points, isolated points, and boundary
points of the set E = {x : x
2 ≥ 2}.
1. The encrypted version of a message is DSWO PYB PEX. If it was encrypted using the function f(p) = (p + 10) mod 26 (the Caesar cipher), what was the original message?
1. Use the Euclidean algorithm to find the greatest common divisor (gcd) of each pairs of integers. Show the steps of calculation.
a) (900,140)
b) (128,729)
1. Suppose a phrase-structure grammar has productions S ® 1S0, S ® 0A, A ® 0. Find derivations of :
a) 00
b) 1000
c) 110000
All of the previous questions used what we know as Euclidean geometry.
There are other geometries. One such is spherical geometry.
What can you say about the sum of the interior angles of a triangle in spherical geometry?
This question is concerning the completion of the square by using plane geometry
The area of a square of length x is : x
2
The area of a rectangle of sides x and y is : xy
Consider the expression : x
2 + 10x
Such can be considered as the sum of two areas viz square x
2 and a rectangle 10x
Now do the following (sketch essential)
Divide the rectangle into two equal areas viz 5x and 5x
Next to the square place the two rectangles with the side x in common. One to the RHS and the other below.
You will now have a ’nearly complete’ square
a) What is the length of each side of the ’missing square’ in your diagram
b) If you add the area of the missing square to make a big square, what is the area of the complete (big) square
c) By using the areas from your sketch, find the expressions ? and ?? in the following:
x
2 + 10x + (?)2 = (??)
The average height of students in a freshman class of a certain school has been 161.27 cm with a population standard deviation of 6.5 cm. Is there a reason to believe that there has been a change in the average height if a random sample of 45 students in the present freshman class has an average height of 150.6 cm? Use a 0.05 level of significance.
Let X be a random variable with the following probability distribution:
x. -3 , 6 , 9
P(X=x). 1/6 , 1/2 , 1/3
Find ug(x), where g(X)= (2X+1)^2
Assume your run the statistics to test the hypothesis and your result is 0.06, and then what will be your decision as a researcher, at P is 5%?
Chinovac Labs tried a new vaccine on 165 randomly selected individuals. From the experiment, it was determined that 82 of them developed immunity. Calculate a 95% confidence interval for the proportion p of individuals in the population for whom the vaccine would help. What is the lower bound of this interval?
